Ultrasound transmission spectroscopy uses measurements of the phase velocity (the sound speed) and attenuation as function of frequency to characterize a medium (e.g. a liquid). Applied to a medium containing suspended particles that affect ultrasound transmission through the medium, this may be used to determine statistical properties of the collection of particles, such as particle density, particle size distribution etc using inverse modelling.
Inverse modelling algorithms for determining particle size distribution from ultrasound speed and/or attenuation are known per se. Inverse modelling involves use of a model (also called the forward model) for predicting measured values as a function of values of parameters of the model. An inverse modelling algorithm uses actually measured values as input to estimate the values of the parameters that correspond to that input. Any model may be used, as long as it predicts ultrasound speed and/or attenuation with a reasonable accuracy given a particle size distribution. In a conventional model of ultrasound transmission by a medium with suspended particles, a low particle concentration approximation of the ultrasound frequency dependent ultrasound propagation speed c(f) of a medium may be expressed asc(f)=c0+integral of N(d)*C1(f,d)
In the medium, the difference in ultrasound phase between points at distance z is 2*PI*z*f/c(f). Herein “c0” is a base value corresponding to the medium in the absence of particles. The second term represents the effect of the particles. The second term can be expressed as an integral over different particle sizes “d” of a product of the number of particles N(d) of particles with a size “d” per unit volume (also called the volume fraction) times a function C1(f, d) that expresses the change of ultrasound speed at the ultrasound frequency “f” due to a unit volume fraction of particles with size d.
The function C1(f, d) may be determined in advance, for example by theoretical prediction, or by measuring ultrasound speed as a function of frequency when a known concentration at a concentration d is present in the medium. Function values for different f, d values may be provided stored in a look up table. In the case of high particle numbers, one or more additional terms corresponding to multiple scattering events, which are non-linear in N(d), may be added in the expression for c(f).
Similarly, the ultrasound frequency dependent ultrasound attenuation a(f) of a medium may be expressed asa(f)=a0+integral of N(d)*A1(f,d)
Herein the function A1(f, d) expresses the change of attenuation at the ultrasound frequency “f” due to a unit concentration of particles with size d. Like C1(f,d), A1(f,d) may be determined in advance. In the medium, the decrease in ultrasound amplitude between points at distance z is exp(−a(f)*z).
Conventionally, inverse modelling of the particle size distribution determines an estimate of N(d) as a function of particle size “d” given measured ultrasound frequency dependent measurements of c(f) and
Usually, N(d) itself is estimated as a parameterized function, for example as a normal distribution, or sum of normal distributions of which the mean, standard deviation and amplitude are estimated given the measurements of c(f) and a(f).
The basic set-up of such a transmission spectroscopy measurement comprises a measurement cell containing the medium and an ultrasound transmitter and receiver on opposite walls of the cell. Effects due to misalignment or differences in equipment behaviour can have a profound impact on the result when the frequency dependent variations in the sound speed due to the particles are quite small. This is especially important at low particle concentrations where there is less measurable variation in the sound speed. To measure very small particles, the upper frequency limit of the system needs to be increased—up to a few hundred MHz. At these frequencies the wavelength is very small, and therefore the demands for the system and electronics design strongly increase as well.
US2004139792 discloses use of ultrasound to determine component ratios and particle sizes of particles in a suspension. Ultrasound attenuation and phase delay of ultrasound transmission through the suspension are determined. The document notes that the attenuation coefficient of a material can be expressed as a logarithm of a ratio of the magnitude of received and transmitted signal spectra, divided by the distance over which attenuation takes place.
To protect the ultrasound transducers of US2004139792 they are separated by thick walls from the suspension. These walls give rise to reflections that may interfere with reception. To minimize these echoes US2004139792 tilts the transducers with respect to the centerline between them, and wedge shaped walls are used. Furthermore, a calibration procedure for a correction for the walls is disclosed. In the calibration procedure wall thickness is measured from the round trip time of ultrasound that reflects on the inner surface of the wall. Similarly, the attenuation coefficient for the walls is calculated from the attenuation of the round trip wave. Signals obtained after multiple reflections may be used for this.